Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions
Salvador Hernández
TL;DR
The paper defines almost periodic functions $AP(U)$ for arbitrary topological structures via translates, without invoking a compactification, and shows $AP(U)$ is a closed commutative Banach subalgebra of $C_{ ext{infty}}(A)$. Its Gelfand space $bU$ serves as the Bohr compactification, with dense evaluation map $\delta:A\to bU$ and an isometric Gelfand transform of $AP(U)$ into $C(bU)$, making $bU$ the maximal compatible compactification. It proves that every basic operation extends to a continuous operation on the Bohr compactification and uses this to realize $bU$ as canonical for arbitrary structures. The results also yield a representation of isometries between spaces of almost periodic functions, showing that non-vanishing isometries have a rigid form $ (Tf)(y)=w(y) f(h(y))$ and, under extra algebraic assumptions, induce isomorphisms between the underlying structures. Together, these findings provide a functional-analytic realization of the Bohr compactification for general topological structures and inform the structure of isometries on almost periodic spaces.
Abstract
The Bohr compactification is a well known construction for (topological) groups and semigroups. Recently, this notion has been investigated for arbitrary structures in \cite{har_kun:bohr_discrete} where the Bohr compactification is defined, using a set-theoretical approach, as the maximal compactification which is compatible with the structure involved. Here, we give a characterization of the continuous functions defined on a product space that can be extended continuously to certain compactifications of the product space. As a consequence, the Bohr compactification of an arbitrary topological structure is obtained as the Gelfand space of the commutative Banach algebra of all almost periodic functions. Previously, almost periodic functions $f$ are defined in terms of translates of $f$ with no reference to any compactification of the underlying structure. An application is given to the representation of isometries defined between spaces of almost periodic functions.